DC power flow in rectangular coordinates Ross Baldick The University of Texas, Austin, TX 78712 baldick@ece.utexas.edu February 20, 2013 1 of 21 Title Page ◀◀ ▶▶ ◀ ▶ Go Back Full Screen Close Quit
Abstract “DC power flow” is an analogy between approximations to the real power components of the power flow equations and a direct current resistive circuit. It can also be interpreted as linearization of the real power components expressed in terms of phasor voltage magnitude and phase, linearized about a “flat start.” The accuracy of DC power flow for estimating real power flow is surprisingly good in many cases, although it has large errors in some cases. We explore linearization of both the real and reactive power equations expressed in terms of real and imaginary parts of the voltage phasor. We focus on linearization about a flat start, which in rectangular coordinates has the voltage phasors each with real part one per unit and imaginary part zero. The resulting approximation has relatively good performance for real power. Because of the analog with linearization in terms of polar voltage representation, we call this approximation “DC power flow in rectangular coordinates.” We also exhibit an exact solution to the power flow equations for the particular case of a lossless network. Keywords DC power flow, linearization. 2 of 21 Title Page ◀◀ ▶▶ ◀ ▶ Go Back Full Screen Close Quit
1 Introduction ∙ Exact or approximate solution of the power flow equations is essential to the planning, operation, and control of power systems. ∙ DC power flow [1, § 4.1.4][2] is an analogy between an approximation to the real power components of the AC power flow equations and a related direct current circuit: – can be interpreted as linearization of the real power components in terms of phasor voltage magnitude and phase, that is, “polar coordinates,” – linearization is about a “flat start,” where the voltage phasors each have magnitude one per unit and angle zero [3]. ∙ Most day-ahead locational electricity markets use DC power flow [4] or linearization. 3 of 21 Title Page ◀◀ ▶▶ ◀ ▶ Go Back Full Screen Close Quit
1.1 Accuracy of DC power flow using polar coordinates ∙ DC power flow is accurate for real power flow, except when: – angle differences across lines are large, or – voltage magnitudes deviate significantly from one per unit [3, 5, 6]. ∙ Precisely the most important conditions for evaluating limits, particularly post-contingency limits! ∙ Linearization for reactive power in terms of polar coordinates is poor. 4 of 21 Title Page ◀◀ ▶▶ ◀ ▶ Go Back Full Screen Close Quit
1.2 Rectangular coordinates ∙ Recent significant progress in power flow and optimal power flow (OPF) has used phasor voltage real and imaginary parts: – “rectangular coordinates” [7, 8, 9]. ∙ Semi-definite programming formulations of the OPF problem using rectangular coordinates have provided provably optimal solutions [10]: – may become the dominant approach to OPF. ∙ Linearization still standard currently in online applications. ∙ Consider linearization in rectangular coordinates. 5 of 21 Title Page ◀◀ ▶▶ ◀ ▶ Go Back Full Screen Close Quit
2 Formulation ∙ v ∈ ℂ n is the vector of complex phasor voltages at all n buses in the transmission system, ∙ i ∈ ℂ n is the vector of complex current injections into the transmission system, ∙ s ∈ ℂ n is the vector of complex power injections into the transmission system. ∙ Represent v in rectangular coordinates by writing: v = 1 + e + j f , where: 1 is the vector of all ones, e , f , ∈ ℝ n , and j is the square root of minus 1. ∙ Define a “flat start” to be v = 1 , corresponding to e = f = 0 . 6 of 21 Title Page ◀◀ ▶▶ ◀ ▶ Go Back Full Screen Close Quit
Formulation, continued ∙ Y = G + jB is the bus admittance matrix for the system: – Off-diagonal entries are equal to minus the admittance of the corresponding series elements joining corresponding buses, – Diagonal entries are the sum of the admittances joined to the corresponding buses, due to both series and shunt elements. G = G series + G shunt , B = B series + B shunt , where: G shunt and B shunt are diagonal, G series and B series are symmetric (putting aside cases such as where transformers have off-nominal turns ratios), and G series 1 = B series 1 = 0 , where 0 is the vector (or depending on interpretation, matrix) of all zeros. – Assume G shunt = 0 . 7 of 21 Title Page ◀◀ ▶▶ ◀ ▶ Go Back Full Screen Close Quit
Formulation, continued ∙ Kirchhoff’s current law: i = Yv , = ( G + jB )( 1 + e + j f ) , = ( G series + G shunt + j ( B series + B shunt ))( 1 + e + j f ) , = ( G series + j ( B series + B shunt ))( 1 + e + j f ) , since G shunt = 0 , = G series 1 + G series e − ( B series + B shunt ) f + j ( G series f + B series 1 + B series e + B shunt 1 + B shunt e ) , = G series e − ( B series + B shunt ) f + j ( G series f + B series e + B shunt 1 + B shunt e ) , ∙ since G series 1 = B series 1 = 0 . ∙ Define superscript † to mean transpose, superscript ‡ to mean Hermitian transpose (that is, transpose of complex conjugate), and diag ( ∙ ) to be a vector consisting of the diagonal elements of its argument. ∙ For any vector x and diagonal matrix D , diag 1 x † ) = x and ( ( x ( D 1 ) † ) = Dx . diag 8 of 21 Title Page ◀◀ ▶▶ ◀ ▶ Go Back Full Screen Close Quit
Formulation, continued ∙ Then: s = diag vi ‡ ) ( , ( ) † ) G series e − ( B series + B shunt ) f ( ( 1 + e + j f ) = diag . G series f + B series e + B shunt 1 + B shunt e − j ( ) ∙ Linearization of s about a flat start of v = 1 preserves the affine terms and discards the purely quadratic terms: ( ) † ) G series e − ( B series + B shunt ) f ( s ≈ diag 1 G series f + B series e + B shunt 1 + B shunt e − j ( ) ( ) † ) ( e + j f ) − jB shunt 1 ( + diag , = G series e − ( B series + B shunt ) f − j ( G series f + B series e + B shunt 1 + B shunt e ) − jB shunt ( e + j f ) , since B shunt is diagonal, = G series e − B series f + j ( − ( B series + 2 B shunt ) e − G series f − jB shunt 1 ) . 9 of 21 Title Page ◀◀ ▶▶ ◀ ▶ Go Back Full Screen Close Quit
Formulation, continued ∙ Separating s into real and reactive power injections: p G series − B series e [ ] [ ][ ] [ ] 0 ≈ − . (1) − ( B series + 2 B shunt ) − G series q f B shunt 1 ∙ Similar form to case in polar coordinates. 10 of 21 Title Page ◀◀ ▶▶ ◀ ▶ Go Back Full Screen Close Quit
3 Special case ∙ Purely quadratic terms in s are: G series f + B series e + B shunt e ( )) † ) G series e − ( B series + B shunt ) f − j ( e + j f ) ( ( , diag ∙ which has real part: ) † + f G series f + B series e + B shunt e ( ) † ) G series e − ( B series + B shunt ) f e ( ( , diag ∙ Following Zhang and Tse [9, Appendix], if e = 0 and G series = 0 then the real part of the quadratic term equals zero. ∙ For given p , if we solve p = − B series f for f then v = 1 + j f is an exact solution to the power flow equations. ∙ Corresponding reactive power injections are: G series f + B shunt 1 ) † − f ( ( B series + B shunt ) f ) † ) q = diag ( ( , − 1 = − G series f − B shunt 1 − diag ( ) † ) ( B series + B shunt ) f f ( 11 of 21 Title Page ◀◀ ▶▶ ◀ ▶ Go Back Full Screen Close Quit
Special case, continued ∙ Analogous to the DC approximation in polar coordinates where the relationship between power and angle is derived under the assumption that the voltage magnitudes are fixed, with the values of reactive injection implicitly determined: – however, DC approximation in polar coordinates is not exact even if G series = 0 . ∙ In contrast, − B series f = p , (2) q = − G series f − B shunt 1 − diag ( ( B series + B shunt ) f ) † ) f ( , (3) ∙ are an exact solution to the power flow equations in the lossless case. 12 of 21 Title Page ◀◀ ▶▶ ◀ ▶ Go Back Full Screen Close Quit
4 Numerical example ∙ Consider a two bus system with: – a single line joining the two buses having series admittance 1 − j 10 and no shunt admittance, – one of the buses having phasor voltage held at 1 + j 0 ∈ ℂ , and – the other bus having phasor voltage 1 + e + j f ∈ ℂ . ∙ We consider the exact and linearized approximations for the real and reactive power injected at the other bus as a function of the bus phasor voltage. ∙ Consider DC approximations in terms of both polar and rectangular coordinates. 13 of 21 Title Page ◀◀ ▶▶ ◀ ▶ Go Back Full Screen Close Quit
4.1 Real power Approximation in polar coordinates 10 p 5 0 −5 Approximation in polar coordinates −10 Approximation in rectangular coordinates 1 f Exact injection 0.5 1 1 + e 0.8 0 0.6 −0.5 0.4 0.2 −1 0 Fig. 1. Real power injection p versus 1 + e + j f . 14 of 21 Title Page ◀◀ ▶▶ ◀ ▶ Go Back Full Screen Close Quit
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